Kinematics-Rejhan Mehmedovic

Kinematics Scalars and Vectors Vectors

Physics is a mathematical science. The underlying concepts and principles have a mathematical  basis. Throughout the course of our study of physics, we will encounter a variety of concepts that have a mathematical basis associated with them. While our emphasis will often be upon the conce concept ptual ual natu nature re of physi physics cs,, we will will give give cons consid ider erab able le and persi persist sten entt atte attent ntio ion n to its its mathematical aspect. The motion of objects can be described by words. Even a person without a background in  physics has a collection of words that can be used to describe moving objects. Words Words and phrases such such as going going fast, fast, stopped stopped,, slowin slowing g down, down, speedi speeding ng up, and turnin turning g provid providee a suffi sufficie cient nt vocabulary for describing the motion of objects. In physics, we use these words and many more. We will will be epandi epanding ng upon this this vocabul vocabulary ary list list with with words words such such as distan distance, ce, displa displacem cement ent,, speed, speed, veloci velocity ty,, and accele accelerat ration ion.. !s we will will soon soon see, see, these these words words are associ associate ated d with with mathematical "uantities that have strict definitions. The mathematical "uantities that are used to describe the motion of objects can be divided into two categories. The "uantity is either a vector  or a scalar scalar.. These These two categor categories ies can be disti distingui nguishe shed d from from one anothe anotherr by their their distin distinct ct definitions# $calars are "uantities that are fully described by a magnitude %or numerical value& alone.  (ectors are "uantities that are fully described by both a magnitude and a direction. d irection. '  (ectors The remainder of this lesson will focus on several eamples of vector and scalar "uantities %distance, %distance, displacement, displacement, speed, velocity, velocity, and acceleratio acceleration&. n&. !s you proceed proceed through through the lesson, lesson, give careful attention to the vector and scalar nature of each "uantity. !s we proceed through other units at The Physics )lassroom Tutorial and become introduced to new mathematical "uantities, the discussion will often begin by identifying the new "uantity as being either a vector  or a scalar.

*istance and *isplacement

*istance and displacement are two "uantities that may seem to mean the same thing yet have distinctly different definitions and meanings.

*istance is a scalar "uantity that refers to +how much ground an object has ' covered+ during its motion. *isplacement is a vector "uantity that refers to +how far out of place an object ' is+ it is the object-s overall change in position.To test your understanding of this distinction, consider the motion depicted in the diagram below. ! physics teacher 

walks  meters East, / meters $outh,  meters West, and finally / meters 0orth.

Even though the physics teacher has walked a total distance of 1/ meters, her displacement is 2 meters. *uring the course of her motion, she has +covered 1/ meters of ground+ %distance 3 1/ m&. 4et when she is finished walking, she is not +out of place+ 5 i.e., there is no displacement for  her motion %displacement 3 2 m&. *isplacement, being a vector "uantity, must give attention to direction. The  meters east cancels the  meters west and the / meters south cancels the / meters north. (ector "uantities such as displacement are direction aware. $calar "uantities such as distance are ignorant of direction. In determining the overall distance traveled by the physics teachers, the various directions of motion can be ignored.  0ow consider another eample. The diagram below shows the position of a cross5country skier  at various times. !t each of the indicated times, the skier turns around and reverses the direction of travel. In other words, the skier moves from ! to 6 to ) to *. Speed and Velocity

7ust as distance and displacement have distinctly different meanings %despite their similarities&, so do speed and velocity. $peed is a scalar "uantity that refers to +how fast an object is moving.+ $peed can be thought of as the rate at which an object covers distance. ! fast5moving object has a high speed and covers a relatively large distance in a short amount of time. ! slow5moving object has a low speed and covers a relatively small amount of distance in a short amount of  time. !n object with no movement at all has a 8ero speed. (elocity is a vector "uantity that refers to +the rate at which an object changes its position.+ Imagine a person moving rapidly 5 one step forward and one step back 5 always returning to the original starting position. While this might result in a fren8y of activity, it would result in a 8ero velocity. 6ecause the person always returns to the original position, the motion would never  result in a change in position. $ince velocity is defined as the rate at which the position changes, this motion results in 8ero velocity. If a person in motion wishes to maimi8e their velocity, then that person must make every effort to maimi8e the amount that they are displaced from their  original position. Every step must go into moving that person further from where he or she started. 9or certain, the person should never change directions and begin to return to the starting  position.(elocity is a vector "uantity. !s such, velocity is direction aware. When evaluating the velocity of an object, one must keep track of direction. It would not be enough to say that an object has a velocity of :: mi;hr. ather than the speed5o5meter maintaining a steady reading, the needle constantly moves up and down to reflect the stopping and starting and the accelerating and decelerating. ead on. ^ Animation

Calculating Average Speed and Average Velocity

The average speed during the course of a motion is often computed using the following formula# Distance Traveled Average Speed = -----------------------Time of Travel

In contrast, the average velocity is often c omputed using this formula A position displacement Average Velocity'=--------------- =----------------time time

?et-s begin implementing our understanding of these formulas with the following problem# Q !"ile on vacation# $isa Carr traveled a total distance of %%& miles (er trip too) * "ours !"at +as "er average speed,

To compute her average speed, we simply divide the distance of travel by the time of travel.

That was easy@ ?isa )arr averaged a speed of :: miles per hour. $he may not have been traveling at a constant speed of :: mi;hr. $he undoubtedly, was stopped at some instant in time %perhaps for  a bathroom break or for lunch& and she probably was going A: mi;hr at other instants in time. 4et, she averaged a speed of :: miles per hour. The above formula represents a shortcut method of  determining the average speed of an object. Average Speed versus nstantaneous Speed

$ince a moving object often changes its speed during its motion, it is common to distinguish  between the average speed and the instantaneous speed. The distinction is as follows.  Instantaneous $peed 5 the speed at any given instant in time. ' ' !verage $peed 5 the average of all instantaneous speeds found simply by a distance;time ratio. 4ou might think of the instantaneous speed as the speed that the speedometer reads at any given instant in time and the average speed as the average of all the speedometer readings during the course of the trip. $ince the task of averaging speedometer readings would be "uite complicated %and maybe even dangerous&, the average speed is more commonly calculated as the distance;time ratio. Boving objects don-t always travel with erratic and changing speeds. emember  that the displacement refers to the change in position and the velocity is based upon this position change. In this case of the teacher-s motion, there is a position change of 2 meters and thus an average velocity of 2 m;s. ere is another eample similar to what was seen before in the discussion of distance and displacement. The diagram below shows the position of a cross5country skier at various times. !t each of the indicated times, the skier turns around and reverses the direction of travel. In other  words, the skier moves from ! to 6 to ) to *. Acceleration

The final mathematical "uantity discussed in ?esson 1 is acceleration. !n often confused "uantity, acceleration has a meaning much different than the meaning associated with it by sports announcers and other individuals. The definition of acceleration is# !cceleration is a vector "uantity that is defined as the rate at which an object changes its velocity. !n object is accelerating if it is changing its velocity. $ports announcers will occasionally say that a person is accelerating if he;she is Time (elocity moving fast. 4et acceleration has nothing to do with going fast. < s 2 m;s, Wo

! person can be moving very fast and still not be accelerating.

12 m;s. 0o

!cceleration has to do with changing how fast an object is moving. If an object is not changing its velocity, then the object is not accelerating. The data at the right are = s representative of a northward5moving accelerating object. The velocity is changing over the course of time. In fact, the velocity is changing by a constant : s amount 5 12 m;s 5 in each second of time. !nytime an object-s velocity is changing, the object is said to be accelerating it has an acceleration.

/2 m;s. 0o =2 m;s. 0o 2 m;s. 0o :2 m;s. 0o

T"e .eaning of Constant Acceleration

$ometimes an accelerating object will change its velocity by the same amount each second. !s mentioned in the previous paragraph, the data table above show an object changing its velocity by 12 m;s in each consecutive second. This is referred to as a constant acceleration since the velocity is changing by a constant amount each second. !n object with a constant acceleration should not be confused with an object with a constant velocity. *on-t be fooled@ If an object is changing its velocity 5whether by a constant amount or a varying amount 5 then it is an accelerating object. !nd an object with a constant velocity is not accelerating. The data tables below depict motions of objects with a constant acceleration and a changing acceleration. 0ote that each object has a changing velocity. !ccelerating )bjects are )hanging Their (elocity... ...by a constant amount ... orby a changing amount each second ... each second ... Time (elocity Time (elocity %m;s& %m;s& %s& %s& 2 2 2 2  1 1 1 $  / / = = : 1/   F 1A Time !ve. (elocity *uring *istance Traveled Total *istance Traveled from